Simplify; express your answer in exponential form. Assume $y\neq 0, r\neq 0$. $\dfrac{{(y^{-3}r^{-2})^{-2}}}{{(y^{5}r)^{-5}}}$
Solution: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(y^{-3}r^{-2})^{-2} = (y^{-3})^{-2}(r^{-2})^{-2}}$ On the left, we have ${y^{-3}}$ to the exponent ${-2}$ . Now ${-3 \times -2 = 6}$ , so ${(y^{-3})^{-2} = y^{6}}$ Apply the ideas above to simplify the equation. $\dfrac{{(y^{-3}r^{-2})^{-2}}}{{(y^{5}r)^{-5}}} = \dfrac{{y^{6}r^{4}}}{{y^{-25}r^{-5}}}$ Break up the equation by variable and simplify. $\dfrac{{y^{6}r^{4}}}{{y^{-25}r^{-5}}} = \dfrac{{y^{6}}}{{y^{-25}}} \cdot \dfrac{{r^{4}}}{{r^{-5}}} = y^{{6} - {(-25)}} \cdot r^{{4} - {(-5)}} = y^{31}r^{9}$